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Compact Two-State-Variable Second-Order Memristor Model
Sungho Kim , Hee-Dong Kim , * and Sung-Jin Choi *
and temperatures can emulate the resistive switching
behavior with a reasonable accuracy; [ 6–11 ] however, because
such an approach requires too much computational power,
the reported models are not suitable for large-scale circuit
simulations.
Alternatively, several compact memristor models have
been developed [ 12–21 ] in which the current–voltage ( I – V ) rela-
tion depends on a state-variable that can evolve in time when
a current is fl owing through the device. Unfortunately, most
of these models have utilized only a one-state-variable, [ 12–19 ]
and they are typically derived by assuming a particular phys-
ical mechanism for the resistive switching and by fi tting the
experimental data to the equations corresponding to this
mechanism. For example, most models are fi tted by assuming
the modulation of the depletion gap length. This assumption
is certainly a simplifi cation of the actual physical mechanism,
and as a result, these models are still not suffi ciently accu-
rate because other multiple mechanisms are involved in the
RS. [ 22 ] Moreover, these models are either nondynamic and
can only predict the steady-state properties under a DC input
signal, [ 12–18 ] or they are phenomenological without including
a physical mechanism. [ 19 ] Recently, the advanced memristor
model based on a two-state-variable has been proposed,
which relies on the evolution of the conductive fi lament (CF)
in terms of both its diameter and depletion gap length. [ 20,21 ]
While these two-state-variable models reproduced the
A key requirement for using memristors in functional circuits is a predictive physical model to capture the resistive switching behavior, which shall be compact enough to be implemented using a circuit simulator. Although a number of memristor models have been developed, most of these models (i.e., fi rst-order memristor models) have utilized only a one-state-variable. However, such simplifi cation is not adequate for accurate modeling because multiple mechanisms are involved in resistive switching. Here, a two-state-variable based second-order memristor model is presented, which considers the axial drift of the charged vacancies in an applied electric fi eld and the radial vacancy motion caused by the thermophoresis and diffusion. In particular, this model emulates the details of the intrinsic short-term dynamics, such as decay and temporal heat summation, and therefore, it accurately predicts the resistive switching characteristics for both DC and AC input signals.
Memristors
DOI: 10.1002/smll.201600088
Prof. S. Kim, Prof. H. -D. Kim Department of Electrical Engineering Sejong University Seoul 05006 , South Korea E-mail: [email protected]
Prof. S.-J. Choi School of Electrical Engineering Kookmin University Seoul 02707 , South Korea E-mail: [email protected]
1. Introduction
The recent progress in memristive [ 1,2 ] devices is promising
for various applications. [ 3,4 ] The development of such appli-
cations and the utilization of the analog switching properties
of memristive devices [ 5 ] will rely on the availability of accu-
rate predictive device models. In oxide-based memristors,
especially relying on the valence change memory effect, the
principle of resistive switching (RS) is believed to be caused
by ion transports (i.e., oxygen vacancies (V O s)) in the oxide
layer, where regions with high concentrations of accumulated
V O s control the conductivity of the device according to the
history of the applied voltage and current. In principle, the
solving of the coupled continuity equations for ions, currents,
valence change memory effect. The proposed model combines
the axial drift of the oxygen vacancies in an applied electric
fi eld and the radial vacancy movement caused by the thermo-
phoresis and diffusion. In particular, this model illustrates the
details of the short-term dynamics of the temperature under
a fast AC input, which allows for the accurate estimation of a
thermally activated RS in an oxide-based memristor.
2. Second-Order Memristor Model
As mentioned earlier, the RS behavior in oxide-based mem-
ristors is associated with the CF growth and rupture due to
the migration of the oxygen vacancies. [ 25 ] To construct an
analytical model by following the memristor’s theoretical
frame, we simplify the model to a single dominant fi lament.
Figure 1 a schematically illustrates the CF evolution during
the reset and set processes. Here, the RS is caused by the cre-
ation of a depletion gap for the V O s near the electrode during
the reset and by the refi lling of the gap with V O s during the
set. [ 9,10,24 ] The CF is defi ned to be that region in the otherwise
insulating fi lm that contains enough V O s to have a metallic
conductivity. [ 25 ] The CF is approximated here as a cylinder
with a radius r and a length l , and the distance between the
tip of the cylinder and the top electrode (TE) is defi ned as a
depleted gap length g . As a result, the entire device can be
regarded as three serial subdivided parts: the depleted gap,
the CF, and the parasitic series resistance ( R S = 350 Ω), as
shown in Figure 1 b. The currents fl owing through the depleted
gap, CF, and R S are expressed as
sinh{( )/ }gap 0/
b 0I I e V V Vg gTEm= −− ( 1a)
( )/{ /( )}CF b a2I V V l rρ π= − ( 1b)
( )/S a BE SI V V R= − ( 1c)
where ρ is the resistivity of the CF, V TE and V BE are the volt-
ages applied to the top and bottom electrodes (TE and BE),
respectively, and V a and V b are the voltages between the subdi-
vided parts. The fi lament resistivity ρ is calculated based on the
Fuchs–Sondheimer approximation for the nanowire resistivity, [ 25 ]
(1 (3 /4 )(1 ))0 r pρ ρ λ= + − , where ρ 0 is the resistivity of the bulk
tantalum, λ is the electron mean free path, and p is the specularity
factor (i.e., the probability for elastic scattering at the CF surface).
In addition, the current through the depleted gap region can be
modeled as a tunneling current [ 13 ] and exponentially depends on
the gap length g . The fi tting parameters are I 0 , V 0 , and g m . Addi-
tionally, the current continuity conditions require that
gap CF S outI I I I= = = ( 1d)
Equations ( 1c – d) determine the device output current–
input voltage ( I out – V appl ) relations. Specifi cally, the device
small 2016, 12, No. 24, 3320–3326
Figure 1. a) Schematic illustration of the model for the reset and set transitions. In the reset transition, the V O migration toward the top electrode leads to the opening of a depleted gap with an increasing g . In the set transition, the V O injection from the tip of the CF into the depleted gap results in the growth of the CF length (step-1) with an increasing diameter (step-2). b) Equivalent model of the CF with the CF considered to be three subdivided parts: the depleted gap, the fi lament, and the series resistance parts. Metallic conduction dominates in the fi lament and series resistance parts, and the tunneling mechanism governs the conduction in the depleted gap region. c) Schematics of V O concentration ( n ) and its gradient in the vertical (d n /d y ) and lateral (d n /d x ) directions.
respectively) followed by small, nonperturbative read voltage
pulses (0.2 V, 1 ms) in intervals. Figure 4 b shows the meas-
ured resistance changes according to the applied pulse trains.
A gradual transition in both the set and reset responses is
clearly observed as the number of applied pulses increases.
The simulated results for this switching behavior match well
with the measured data for both the set and reset pulse trains.
Therefore, this simulation can describe the AC resistive
switching response reliably in addition to
the DC characteristics.
To further evaluate the model,
Figure 4 c shows the calculated g and r
during consecutive pulse trains. During
the reset process, J drift is increased due
to the high electric fi eld at the narrow
depleted gap, and the high temperature
caused by the high reset current leads to
simultaneous increases in both J Fick and
J Soret . An increase in g and a decrease in
r happen simultaneously because J Soret
and J drift , respectively, are more domi-
nant than J Fick . However, in the set pro-
cesses, the temperature inside the CF is
lower than that reached during the reset
pulse because the Joule heating does not
proceed effectively while the strong elec-
tric fi eld at the depleted gap increases
J drift , and this leads to a faster connection
between the tip of the CF and the TE.
Consequently, the depleted gap is pre-
dominantly refi lled by the V O migration
due to the J drift (here, both J drift and J Fick
have the same direction). Interestingly,
after the CF reconnects, the reconnected
CF allows the current fl ow to generate
Joule heating, which leads to an increase in J Fick . While the
temperature during the set process is lower than that of the
reset process as shown in Figure 3 b, J Soret is also increased
but temperature during the set process is lower than that of
the reset process as shown in Figure 3 b. As a result, J Fick is
more dominant than J Soret , and r is increased by J Fick during
the subsequent set pulses. Accordingly, the CF change is a
two-step process during the set train: (1) fast vertical gap
small 2016, 12, No. 24, 3320–3326
Figure 3. a) Measured and calculated DC I – V characteristics of the Pt/Ta 2 O 5 /TaO x /Pt device. The measured device size is 5 µm × 5 µm, and the voltage sweep speed is 2 V s −1 . b) Calculated parameter changes as a function of time during the DC sweep.
Figure 4. a) Schematics of the applied pulse trains used for the measurement of the AC switching behavior. Each pulse train consists of 20 set or reset pulses (−0.9 and 1.1 V, respectively; 10 µs) followed by small, nonperturbative read voltage pulses (0.2 V, 1 ms) in intervals. b) Measured and calculated switching behavior. c) Calculated g and r at different points in the consecutive pulse trains.
fi lling by the fi eld-driven migration of the internal V O s and
( 2) the following lateral expansion of the CF by diffusion.
Therefore, a relatively abrupt transition in the device resist-
ance occurs during the set because of the faster connection
of the tip of the CF and the TE. From these results, it can be
inferred that the balance among the fl uxes and the internal
temperature dynamics are a crucial factor for governing the
resistive switching, and the proposed model accurately rep-
resents this switching process.
As mentioned above, the short-term temperature
dynamics (i.e., the second-order memristor effect) plays an
important role in governing the RS of the device. To sys-
tematically study how the RS behavior is affected by the
internal temperature dynamics, the transient response of the
RS was measured at different t SET and t interval confi gurations
(the measurement procedure is the same as the previously
reported one [ 24 ] ), as shown in Figure 5 a. In this measurement,
a train of set pulses were applied with the pulse amplitude
fi xed at V SET and t SET while t interval ranged from 100 ns to 1 µs.
Figure 5 b shows that two qualitatively different RS behaviors
can be observed depending on t interval : (1) when t interval is short
(100 ns), abrupt RS behavior is observed because the heat
generated by the set pulses can be temporarily accumulated
due to the short t interval ; ( 2) under a long t interval (1 µs), each
set pulse does not create a high enough temperature rise
(whereas the heat generated during the previous set pulses
will be effectively dissipated during the long interval), and
therefore, heat accumulation is not achieved, resulting in a
slower V O migration and a more gradual RS. Our second-
order memristor model captures the nature of the short-
term temperature dynamics accurately. Figure 5 c shows the
calculated transient response of I out , which is consistent with
the measured data. The model can emulate the different RS
behavior according to t i nterval , where the internal tempera-
ture dynamics are also accurately predicted by the model, as
shown in Figure 5 d. Figure 5 d shows the calculated transient
response of T CF and T bulk . Due to the temporal heat accumu-
lated with a short t i nterval , T CF exceeds 650 K, and T bulk is also
increased to near 400 K. In contrast, T CF is maintained ≈500 K
under a long t i nterval , with an almost constant T bulk at 300 K.
small 2016, 12, No. 24, 3320–3326
Figure 5. a) Schematic of pulse trains used during the measurements. A pulse train consists of a single reset pulse ( V RESET = 1.4 V, pulse duration t RESET = 40 µs) and 100 subsequent set pulses ( V SET = −0.9 V, pulse duration t SET = 100 ns), where the intervals between the set pulses ( t interval ) are either 1 µs or 100 ns. b) Measured transient responses with different t interval s, 1 µs and 100 ns. An abrupt RS with a t interval = 100 ns (left) and a gradual RS with a t interval = 1 µs (right) was observed. c) Simulated transient responses with different t interval s. d) Transient response of T CF and T bulk obtained from the analytical second-order memristor model.
The short-term dynamics of temperature and the temporal
summation effect allow for a different memristor resistance
change. As a result, this two-state-variable second-order
memristor model, with an emphasis on the internal dynamics,
enables the accurate prediction of the dynamic behavior of
the device, where the fi rst state-variable is the device resist-
ance determined by g and r , and the second state-variable is
given by T CF and T bulk .
4. Conclusion
In summary, we developed a two-state-variable second-
order memristor model and fi tted it using our experimental
measurement data in Pt/Ta 2 O 5 /TaO x /Pt memristor devices.
The model reproduced the experimental data accurately,
especially the transient response under the applying pulses.
The two-state variables determine the device resistance. The
depleted gap length g and the radius of the conducting fi la-
ment r were responsible directly for the change of the device
resistance (i.e., the fi rst state-variable). In addition, the unre-
vealed internal temperature dynamics described by T CF and
T bulk were also responsible for the shorter timescale system-
atic changes in the device resistance. This second-order effect
is important in understanding fast switching, and this com-
pact model is suitable for fast circuit simulations because of
the simple form of its equations. This study provided a useful
methodology to exploit second-order memristors, and it will
help to fi nd other applications of second-order memristors in
the future.
5. Experimental Section
Transient Response Measurements : For the transient meas-urements, the device was serially connected to a series resistor (50 Ω), which was used to read out the transient current through the device. A designed pulse train was applied to the device’s TE. A digital oscilloscope (with both input resistances of CH1 and CH2 set to 1 MΩ) was used to record the voltage transients at the TE and BE locations in the circuit. The net applied voltage and current in the device were calculated as V appl = V CH1 − V CH2 and I out = V CH2 /50 Ω, respectively.
Acknowledgements
The work was supported by the National Research Foun-dation of Korea through the Ministry of Education,
Science and Technology, Korean Government, under Grant 2013R1A1A1057870.
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Received: January 11, 2016 Revised: March 31, 2016 Published online: May 6, 2016